arrow
Volume 13, Issue 3
Effect of Space Dimensions on Equilibrium Solutions of Cahn-Hilliard and Conservative Allen-Cahn Equations

Hyun Geun Lee, Junxiang Yang, Junseok Kim & Jintae Park

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 644-664.

Published online: 2020-03

Export citation
  • Abstract

In this study, we investigate the effect of space dimensions on the equilibrium solutions of the Cahn-Hilliard (CH) and conservative Allen-Cahn (CAC) equations in one, two, and three dimensions. The CH and CAC equations are fourth-order parabolic partial and second-order integro-partial differential equations, respectively. The former is used to model phase separation in binary mixtures, and the latter is used to model mean curvature flow with conserved mass. Both equations have been used for modeling various interface problems. To study the space-dimension effect on both the equations, we consider the equilibrium solution profiles for symmetric, radially symmetric, and spherically symmetric drop shapes. We highlight the different dynamics obtained from the CH and CAC equations. In particular, we find that there is a large difference between the solutions obtained from these equations in three-dimensional space.

  • AMS Subject Headings

65M55, 65N06, 76D05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

cfdkim@korea.ac.kr (Junseok Kim)

  • BibTex
  • RIS
  • TXT
@Article{NMTMA-13-644, author = {Lee , Hyun GeunYang , JunxiangKim , Junseok and Park , Jintae}, title = {Effect of Space Dimensions on Equilibrium Solutions of Cahn-Hilliard and Conservative Allen-Cahn Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {3}, pages = {644--664}, abstract = {

In this study, we investigate the effect of space dimensions on the equilibrium solutions of the Cahn-Hilliard (CH) and conservative Allen-Cahn (CAC) equations in one, two, and three dimensions. The CH and CAC equations are fourth-order parabolic partial and second-order integro-partial differential equations, respectively. The former is used to model phase separation in binary mixtures, and the latter is used to model mean curvature flow with conserved mass. Both equations have been used for modeling various interface problems. To study the space-dimension effect on both the equations, we consider the equilibrium solution profiles for symmetric, radially symmetric, and spherically symmetric drop shapes. We highlight the different dynamics obtained from the CH and CAC equations. In particular, we find that there is a large difference between the solutions obtained from these equations in three-dimensional space.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0159}, url = {http://global-sci.org/intro/article_detail/nmtma/15779.html} }
TY - JOUR T1 - Effect of Space Dimensions on Equilibrium Solutions of Cahn-Hilliard and Conservative Allen-Cahn Equations AU - Lee , Hyun Geun AU - Yang , Junxiang AU - Kim , Junseok AU - Park , Jintae JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 644 EP - 664 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0159 UR - https://global-sci.org/intro/article_detail/nmtma/15779.html KW - Cahn-Hilliard equation, conservative Allen-Cahn equation, equilibrium solution, finite difference method, multigrid method. AB -

In this study, we investigate the effect of space dimensions on the equilibrium solutions of the Cahn-Hilliard (CH) and conservative Allen-Cahn (CAC) equations in one, two, and three dimensions. The CH and CAC equations are fourth-order parabolic partial and second-order integro-partial differential equations, respectively. The former is used to model phase separation in binary mixtures, and the latter is used to model mean curvature flow with conserved mass. Both equations have been used for modeling various interface problems. To study the space-dimension effect on both the equations, we consider the equilibrium solution profiles for symmetric, radially symmetric, and spherically symmetric drop shapes. We highlight the different dynamics obtained from the CH and CAC equations. In particular, we find that there is a large difference between the solutions obtained from these equations in three-dimensional space.

Hyun Geun Lee, Junxiang Yang, Junseok Kim & Jintae Park. (2020). Effect of Space Dimensions on Equilibrium Solutions of Cahn-Hilliard and Conservative Allen-Cahn Equations. Numerical Mathematics: Theory, Methods and Applications. 13 (3). 644-664. doi:10.4208/nmtma.OA-2019-0159
Copy to clipboard
The citation has been copied to your clipboard